Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

492 CHAPTER 16 Shear of Beams

whichgives

∫s

0

qs

Gt

ds=(ws−w 0 )+ 2 AOs

dθ

dz

+

du

dz

(xs−x 0 )+

dv

dz

(ys−y 0 ), (16.21)

whereAOsis the area swept out by a generator, center at the origin of axes, O, from the origin fors
toanypointsaroundthecrosssection.Continuingtheintegrationcompletelyaroundthecrosssection
yields,fromEq.(16.21) ∮
qs
Gt

ds= 2 A

dθ

dz

fromwhich

dθ

dz

=

1

2 A

∮

qs

Gt

ds (16.22)

SubstitutingfortherateoftwistinEq.(16.21)fromEq.(16.22)andrearranging,weobtainthewarping
distributionaroundthecrosssection

ws−w 0 =

∫s

0

qs

Gt

ds−

AOs

A

∮

qs

Gt

ds−

du

dz

(xs−x 0 )−

dv

dz

(ys−y 0 ) (16.23)

UsingEqs.(16.11)toreplacedu/dzanddv/dzinEq.(16.23),wehave

ws−w 0 =

∫s

0

qs

Gt

ds−

AOs

A

∮

qs

Gt

ds−yR

dθ

dz

(xs−x 0 )+xR

dθ

dz

(ys−y 0 ) (16.24)

ThelasttwotermsinEq.(16.24)representtheeffectofrelatingthewarpingdisplacementtoanarbitrary
origin,whichitselfsuffersaxialdisplacementduetowarping.Inthecasewheretheorigincoincides
withthecenteroftwistRofthesection,thenEq.(16.24)simplifiesto

ws−w 0 =

∫s

0

qs

Gt

ds−

AOs

A

∮

qs

Gt

ds (16.25)

Inproblemsinvolvingsinglyordoublysymmetricalsections,theoriginforsmaybetakentocoincide
withapointofzerowarpingwhichwilloccurwhereanaxisofsymmetryandthewallofthesection
intersect.Forunsymmetricalsections,theoriginforsmaybechosenarbitrarily.Theresultingwarping
distributionwillhaveexactlythesameformastheactualdistributionbutwillbedisplacedaxiallyby
the unknown warping displacement at the origin fors. This value may be found by referring to the
torsionofclosedsectionbeamssubjecttoaxialconstraint.Intheanalysisofsuchbeams,itisassumed
thatthedirectstressdistributionsetupbytheconstraintisdirectlyproportionaltothefreewarpingof
thesection—thatis,

σ=constant×w